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Mirrors > Home > MPE Home > Th. List > fmptapd | Structured version Visualization version GIF version |
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
Ref | Expression |
---|---|
fmptapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fmptapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fmptapd.s | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
fmptapd.c | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptapd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
2 | fmptapd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fmptapd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 1, 2, 3 | fmptsnd 7162 | . . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
5 | 4 | uneq2d 4158 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
6 | mptun 6689 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
8 | fmptapd.s | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
9 | 8 | mpteq1d 5236 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
10 | 5, 7, 9 | 3eqtr2d 2772 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 {csn 4623 ⟨cop 4629 ↦ cmpt 5224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-mpt 5225 |
This theorem is referenced by: fmptpr 7165 poimirlem3 37003 poimirlem4 37004 poimirlem16 37016 poimirlem17 37017 poimirlem19 37019 poimirlem20 37020 |
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